Algebra and Functions
The Algebra and Functions domain tests symbolic fluency and the ability to connect expressions, equations, inequalities, functions and graphs. Questions commonly require simplifying or factorising, solving equations, interpreting function notation and recognising sequence or variation patterns.
Core rules and concepts
- Simplify expressions by collecting like terms; in rational expressions only common factors may be cancelled, never individual terms.
- For ax + b = c with a ≠ 0, x = (c − b)/a. When solving an inequality, multiplying or dividing by a negative number reverses the inequality sign.
- A quadratic may be solved by factorising, completing the square or the formula x = [−b ± √(b² − 4ac)]/(2a). The discriminant b² − 4ac determines whether the roots are real and distinct, real and equal, or non-real.
- Simultaneous linear equations can be solved by substitution or elimination; the solution must satisfy both equations.
- Function notation f(x) names the output for input x. The domain is the set of allowed inputs; the range is the set of outputs.
- An arithmetic sequence has nth term a + (n − 1)d; a geometric sequence has nth term ar^(n − 1), where r is the common ratio.
Worked example
Solve 2x² − 5x − 3 = 0. The product ac is 2 × (−3) = −6, and two numbers with product −6 and sum −5 are −6 and 1. Split the middle term: 2x² − 6x + x − 3. Factor in pairs: 2x(x − 3) + 1(x − 3) = (2x + 1)(x − 3). Hence x = −1/2 or x = 3. Substituting either value into the original equation verifies the result.
Traps that cost marks
- Reversing an inequality only when it is convenient, rather than whenever a negative multiplier or divisor is used.
- Cancelling x in (x + 3)/x to obtain 3, which is invalid because x is not a factor of the whole numerator.
- Treating f(x + 1) as f(x) + 1; function notation applies the rule to the whole input.
- Forgetting to check a rational equation for values that make a denominator zero.